Simplify the following expression and state the condition under which the simplification is valid. $k = \dfrac{x^2 - 36}{x - 6}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = x$ $ b = \sqrt{36} = -6$ So we can rewrite the expression as: $k = \dfrac{({x} {-6})({x} + {6})} {x - 6} $ We can divide the numerator and denominator by $(x - 6)$ on condition that $x \neq 6$ Therefore $k = x + 6; x \neq 6$